In a right-angled triangle ABC, the hypotenuse AB is 12 cm, and the angle A is 60 degrees.

In a right-angled triangle ABC, the hypotenuse AB is 12 cm, and the angle A is 60 degrees. CD – the height lowered from the top of the right angle C to the hypotenuse AB. Find the length of the line segment AD.

1. We calculate the length of the leg BC of the right-angled triangle ABC through the sine A:

BC: AB = sine ∠A = sine 60 ° = √3 / 2.

BC = AB x √3 / 2 = 12 x √3 / 2 = 6√3 centimeters.

2. AC = √AB² – BC² (by the Pythagorean theorem).

AC = √12² – (6√3) ² = √144 – 108 = √36 = 6 centimeters.

3. Calculate the length of the segment AD, which is the leg of the right-angled triangle ACD, in terms of the cosine ∠A:

AD: AC = cosine ∠A = cosine 60 ° = 1/2.

AD = AC x 1/2 = 6 x 1/2 = 3 centimeters.

Answer: the length of the segment AD is 3 centimeters.